Abstract: New computation algorithms for a fluid-dynamic mathematical model
of flows on networks are proposed, described and
tested.
First we improve the classical Godunov
scheme (G) for a special flux function,
thus obtaining a more efficient method, the Fast Godunov
scheme (FG) which reduces the number of evaluations for the numerical
flux.
Then a new method, namely the Fast Shock Fitting
method (FSF), based on good theorical properties of the solution of the
problem is introduced.
Numerical results and efficiency tests are presented in order to show the
behaviour of FSF in comparison with G, FG and a conservative
scheme of second order.

Abstract: We consider a multiscale model describing the flow
of a concentrated suspension. The model couples the macroscopic equation
of conservation of momentum with a nonlinear nonlocal kinetic equation
describing at the
microscopic level the rheological behaviour of the fluid. We study the
long-time limit of the time-dependent solution. For this
purpose, we use the
entropy method to prove the convergence to equilibrium of the kinetic
equation.

Abstract: It is well known that, in the presence of an attractive force having
a Coulomb singularity, scattering solutions of the nonrelativistic
Abraham--Lorentz--Dirac equation having nonrunaway character do not
exist, for the case of motions on the line. By numerical
computations on the full three dimensional case, we give indications
that indeed there exists a full tube of initial data for which
nonrunay solutions of scatterig type do not exist. We also give a
heuristic argument which allows to estimate the size of such a tube
of initial data. The numerical computations also show that in a thin
region beyond such a tube one has the nonuniqueness phenomenon, i.e.
the "mechanical'' data of position and velocity do not uniquely
determine the nonrunaway trajectory.

Abstract: In this paper we obtain Meyers type regularity estimates for
approximate solutions of
nonlinear
elliptic equations. These estimates are used in the
analysis of a numerical scheme obtained from a numerical homogenization of
nonlinear elliptic equations. Numerical homogenization of nonlinear
elliptic equations results in discretization schemes that require
additional integrability of the approximate solutions.
The latter motivates our work.

Abstract: We study the dispersive evolution of modulated pulses in a
nonlinear oscillator chain embedded in a background field. The atoms
of the chain interact pairwise with an arbitrary but finite number of
neighbors. The pulses are modeled as macroscopic modulations of the
exact spatiotemporally periodic solutions of the linearized model. The
scaling of amplitude, space and time is chosen in such a way that we can
describe how the envelope changes in time due to dispersive effects. By
this multiscale ansatz we find that the macroscopic evolution of the
amplitude is given by the nonlinear Schrödinger equation. The main part
of the work is focused on the justification of the formally derived
equation: We show that solutions which have initially the form of the
assumed ansatz preserve this form over time-intervals with a positive
macroscopic length. The proof is based on a normal-form transformation
constructed in Fourier space, and the results depend on the validity of
suitable nonresonance
conditions.

Abstract: In this work we analyze a Gause type predator-prey model with a
non-monotonic functional response and we show that it has two limit
cycles encircling an unique singularity at the interior of the first
quadrant, the innermost unstable and the outermost stable,
completing the results obtained in previous paper [12, 17, 26, 28].
Moreover, using the Poisson bracket we give a
proof, shorter than the ones found in the literature, for
determining the type of a cusp point of a singularity at the first
quadrant.

Abstract: In this paper, we present a new model for optimal control of discrete event systems (DESs) with
an arbitrary control pattern. Here, a discrete event system is defined as a collection of event
sets that depend on strings. When the system generates a string, the next event that may occur
should be in the corresponding event set. In the optimal control model, there are rewards for
choosing control inputs at strings and the sets of available control inputs also depend on
strings. The performance measure is to find a policy under the condition where the discounted
total reward among strings from the initial state is maximized. By applying ideas from Markov
decision processes, we divide the problem into three sub-cases where the optimal value is
respectively finite, positive infinite and negative infinite. For the case with finite optimal
values, the optimality equation is shown and further characterized with its solutions. We also
characterize the structure of the set of all optimal policies. Moreover, we discuss invariance
and closeness of several languages. We present a new supervisory control problem of DESs with
the control pattern being dependent on strings. We study the problem in both the event feedback
control and the state feedback control by generalizing concepts of invariant and closed
languages/predicates. Finally, we apply the above model and results to a job-matching problem.

Abstract: In this paper, a discrete-time system, derived from a
predator-prey system by Euler's method with step one, is
investigated in the closed first quadrant $R_+^2$. It is shown
that the discrete-time system undergoes fold bifurcation, flip
bifurcation and Neimark-Sacker bifurcation, and the discrete-time
system has a stable invariant cycle in the interior of $R_+^2$
for some parameter values. Numerical simulations are provided to
verify the theoretical analysis and show the complicated dynamical
behavior. These results reveal far richer dynamics of the discrete
model compared with the same type continuous model.

Abstract: This paper proposes a novel neural network model for associative memory
using dynamical systems. The proposed model is based on synthesizing the
external input vector, which is different from the conventional approach
where the design is based on synthesizing the connection matrix. It is shown
that this new neural network (a) stores the desired prototype patterns as
asymptotically stable equilibrium points, (b) has no spurious states, and
(c) has learning and forgetting capabilities. Moreover, new learning and
forgetting algorithms are also developed via a novel operation on the matrix
space. Numerical examples are presented to illustrate the effectiveness of
the proposed neural network for associative memory. Indeed, results of
simulation experiments demonstrate that the neural network is effective and
can be implemented easily.

Abstract: We study in this paper the bifurcation and stability of the
solutions of the Rayleigh-Bénard convection which has the infinite
Prandtl number, using a notion of bifurcation called attractor
bifurcation. We prove that the problem bifurcates from the
trivial solution to an attractor $\A_R$ when the Rayleigh number
$R$ crosses the critical Rayleigh number $R_c$. As a special case,
we also prove another result which corresponds to the classical
pitchfork bifurcation, that this bifurcated attractor $\A_R$
consists of only two stable steady states when the first eigenvalue
$R_1$ is simple.

Abstract: Non-linear difference equation models are employed in biology to
describe the dynamics of certain populations and their interaction
with the environment. In this paper we analyze a non-linear system
describing community intervention in mosquito control through
management of their habitats. The system takes the general form:

where the function $h\in C^{1}$ ( [ $0,\infty$) $\to $ [$0,1$] ) satisfying
certain properties, will denote either $h(t)=h_{1}(t)=e^{-t}$
and/or $h(t)=h_{2}(t)=1/(1+t).$ We give conditions in terms of
parameters for boundedness and stability. This enables us to explore
the dynamics of prevalence/community-activity systems as affected by
the range of parameters.

Abstract: This paper extends Runge-Kutta discontinuous Galerkin (RKDG) methods
to a nonlinear Dirac (NLD) model in relativistic quantum physics,
and investigates interaction dynamics of corresponding solitary
wave solutions. Weak inelastic interaction in
ternary collisions is first observed by using high-order accurate schemes on finer
meshes. A long-lived oscillating state is formed with an approximate constant
frequency in collisions of two standing waves; another is with an increasing frequency
in collisions of two moving solitons.
We also prove three continuum conservation laws of the NLD model and
an entropy inequality, i.e. the total charge non-increasing, of the
semi-discrete RKDG methods, which are demonstrated by various
numerical examples.

Abstract: This paper deals with the behavior of symmetric discrete--time
systems with delays. The influence of the delay over these systems
is analyzed in the stabilization problem. Furthermore, conditions
on the system are given in order to solve the pole--assignment
problem. Finally, some examples are shown with the aim to clarify
the obtained results.